Find the distance between the point ${(2, -4)}$ and the line $\enspace {y = 2x + 7}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Answer: First, find the equation of the perpendicular line that passes through ${(2, -4)}$ The slope of the blue line is ${2}$ , and its negative reciprocal is ${-\dfrac{1}{2}}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = -\dfrac{1}{2}x + b}\thinspace$ We can plug our point, ${(2, -4)}$ , into this equation to solve for ${b}$ , the y-intercept. $-4 = {-\dfrac{1}{2}}(2) + {b}$ $-4 = -1 + {b}$ $-4 + 1 = {b} = -3$ The equation of the perpendicular line is $\enspace {y = -\dfrac{1}{2}x - 3}\thinspace$ We can see from the graph (or by setting the equations equal to one another) that the two lines intersect at the point ${(-4, -1)}$ . Thus, the distance we're looking for is the distance between the two red points. The distance formula tells us that the distance between two points is equal to: $\sqrt{( x_{1} - x_{2} )^2 + ( y_{1} - y_{2} )^2}$ Plugging in our points ${(2, -4)}$ and ${(-4, -1)}$ gives us: $\sqrt{( {2} - {-4} )^2 + ( {-4} - {-1} )^2}$ $= \sqrt{( 6 )^2 + ( -3 )^2} = \sqrt{45} = 3\sqrt{5}$ The distance between the point ${(2, -4)}$ and the line $\thinspace {y = 2x + 7}\enspace$ is $\thinspace3\sqrt{5}$.